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I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field theory (which apparently is false).

There, the author defines a differential $\delta$ on the space $\mathcal{F}$ of trajectories (I don´t know if they´re smooth or if it´s $H^1$, because the first would not be a complete manifold in general) in a given riemmanian manifold $X$ (in this case, $X = \mathbb{R}^n$), then he define a map $x : \mathcal{F} \times \mathbb{R} \longrightarrow X$ that just pick a path and evaluate it at some time $t$. However, he assumes that $\delta d + d \delta = 0$, where $d$ is the differential on the real line. Why do they anticommute?

After this, given the lagrangian $L = \frac{1}{2}m \langle \dot{x}, \dot{x}\rangle dt$ and a functional $S = \int_{t_0}^{t_1} L$. That´s ok, but then, he computes $\delta S$ and assumes that the space of extremal solutions $\mathcal{M} \subset \mathcal{F}$ are the solutios to $\ddot{x} = 0$ (Why?).

After computing these stuffs, he considers the forms $\gamma(t) = m \langle \dot{x}(t), \delta x(t) \rangle$, because of the boundary of the action $S$, which is an object analogous to the Liouville one-form (tautological form), then he defines an analogue to the sympletic form $\omega = \delta \gamma = m \langle \delta \dot{x}, \delta \dot{x} \rangle$ (What are these inner product brackets? Is this just a typo?). Furthermore, after this, he defines a connection $\nabla (t_0)$ in a $\mathbb{R}$ principal bundle over $\mathcal{M}$ called $T(t)$ (How is this bundle?) using $\gamma (t)$ (How?). Then, he says that all these bundles are isomorphic (Why?) and, then, pick the inverse limit of these bundles (probably in $t$) , then he says that this bundle has curvature equals $\omega$ (Why?Furthermore, what is the meaning in picking a filtrant limit of isomorphic things, given by arrows that are isomorphisms?).

Thanks in advance.

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After talking to some friends, I could understand roughly the assumptions that the book was using. The main point is that they assumed a lot of variational bicomplex (main references: Vinogradov, Andersson etc) stuffs without even mentioning about it. –  user40276 Apr 29 at 15:35

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